Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{8t^3 - 16t^2 - 192t}{2t^3 - 12t^2 - 80t}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {8t(t^2 - 2t - 24)} {2t(t^2 - 6t - 40)} $ $ q = \dfrac{8t}{2t} \cdot \dfrac{t^2 - 2t - 24}{t^2 - 6t - 40} $ Simplify: $ q = 4 \cdot \dfrac{t^2 - 2t - 24}{t^2 - 6t - 40}$ Since we are dividing by $t$ , we must remember that $t \neq 0$ Next factor the numerator and denominator. $ q = 4 \cdot \dfrac{(t + 4)(t - 6)}{(t + 4)(t - 10)}$ Assuming $t \neq -4$ , we can cancel the $t + 4$ $ q = 4 \cdot \dfrac{t - 6}{t - 10}$ Therefore: $ q = \dfrac{ 4(t - 6)}{ t - 10 }$, $t \neq -4$, $t \neq 0$